Logic for Gray Code Computation

Gray code is a well-known binary number system such that neighboring values differ in one digit only. Tsuiki (2002) has introduced Gray code to the field of real number computation. He assigns to each number a unique 1⊥-sequence, i.e., a (possibly infinite) sequence of {−1, 1,⊥} such that at most one copy of ⊥ (meaning undefinedness) is contained in the sequence not as the last character. In this paper we take a logical and constructive approach to study real number computation based on Gray code. Instead of Tsuiki’s indeterministic multihead Type-2 machine, we use pre-Gray code, which is a relaxed Gray code, to avoid the difficulty due to ⊥ which prevents sequential access to a stream. We extract real number algorithms from proofs in an appropriate formal theory involving inductive and coinductive definitions. Examples are algorithms transforming pre-Gray code into signed digit representations of real numbers, and conversely, the average for pre-Gray code and a bounded translator from pre-Gray code into Gray code. These examples are formalized in the proof assistant Minlog.